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Applications for Testing Hypothesis
Why do we need two methods if they will always lead to the same decision? Well, when learning about hypothesis tests and becoming comfortable with their logic, many people find the rejection region method a little easier to apply. However, when we start to rely on statistical software for conducting hypothesis tests in later chapters of the book, we will find the p-value method easier to use. At this stage, when doing hypothesis test calculations by hand, it is helpful to use both the rejection region method and the p-value method to reinforce learning of the general concepts. This also provides a useful way to check our calculations since if we reach a different conclusion with each method we will know that we have made a mistake.
Power (1-beta): The ability of a statistical test to detect a real difference when there is one; the probability of correctly rejecting the null hypothesis. Determined by alpha and sample size.
null hypothesis significance test
An alternative way to conduct a hypothesis test is to again assume initially that the null hypothesis is true, but then to calculate the probability of observing a t-statistic as extreme as the one observed or even more extreme (in the direction that favors the alternative hypothesis). This is known as the p-value (sometimes also called the observed significance level):
If the p-value is too "small," then this suggests that it seems unlikely that the null hypothesis could have been true—so we reject it in favor of the alternative. Otherwise, the t-statistic could well have arisen while the null hypothesis held true—so we do not reject it in favor of the alternative. Again, the significance level chosen tells us how small is small: If the p-value is less than the significance level, then reject the null in favor of the alternative; otherwise, do not reject it. For the home prices example:
null hypothesis significance testing procedure.
When frequentists have some rejection scheme then that is fine. What they express is not wether a hypothesis is true or false, or the probability for such cases. They are not able to do that (without priors). What they express instead is something about the failure rate (confidence) of their method (given certain assumptions are true).
One way to get out all of this is to elliminate the concept of probability. If you observe the entire population of 100 marbles in the vase then you can express certain statements about a hypothesis. So, if you become omniscient and the concept of probability is irrelevant, then you can state wether a hypothesis is true or not (although probability is also out of the equation)
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Hypothetical statistics for fever and appendicitis.
Given that prior probabilities for hypotheses in science and medicine are often difficult to specify explicitly in precise numerical terms, does this mean that any prior probability for a hypothesis is as good as any other? There are at least two reasons that this is not the case. First, pragmatically, people do not treat prior probabilities regarding scientific or medical hypotheses as arbitrary. To the contrary, they go to great lengths to bring their probabilities into line with existing evidence, usually by integrating multiple information sources, including direct empirical experience, relevant theory (for example an understanding of physiology), and literature concerning prior work on the hypothesis or related hypotheses. These prior probability assignments help scientists and physicians choose which hypotheses deserve further investment of time and resources. Moreover, while these probability estimates are individualized, this does not imply that each person's 'subjective' estimate is equally valid. Generally, experts with greater knowledge and judgement can be expected to arrive at more intelligent prior probability assignments, that is their assignments can be expected to more closely approximate the probability an 'ideal observer' would arrive at based on optimally processing all of the existing evidence. Second, in a more technical vein, methods for estimating accurate prior probabilities from existing data are an active topic of research, and are likely to lead to increased and more explicit use of 'Bayesian statistics' in the medical literature [, , , , , , , , ].
F-test: A hypothesis test for comparing variances.
The big difference between a study and a clinical test is that there is no real way of knowing how likely or unlikely a hypothesis is a priori. In order to have a predictive value in a clinical test, you need a prevalence or pre-test probability. This does not exist in science. It is the job of the scientist to convince us that the pre-test probability is reasonably high so that a result will be accepted. They do this by laying the scientific groundwork (introduction), laying out careful methods, particularly avoiding bias and confounders (methods), and describing the results carefully. Thereafter, they use the discussion section to outright and unabashedly try to convince us their results are right. But in the end, we do the positive predictive value calculation in our head as we read a paper... As an example, one person reads the SPARCL study and says, 'I do not CARE that the P-value shows statistical significance, it is hooey to say that statins cause intracranial hemorrhage.'... They have set a very low pre-test probability in their head. Another person reads the same study and says, 'I have wondered about this because I have seen lots of bleeds in people on statins.' They have set a much higher pre-test probability.
Conduct a hypothesis test at the .01 significance level.
The answer to the quiz at the beginning of this paper is plain from the preceding discussion. Given a P-value that reaches significance (such that the NHSTP would have us conclude that H1 is true), what conclusions are we actually justified in drawing regarding the probability that either hypothesis H1 or H0 is true? Answers (1), (2), and (5) are incorrect because the NHSTP, which corresponds to the 'hard' version of 'probabilistic proof by contradiction' is an invalid argument. Answers (3), (4), and (6) are invalid because the 'softened' version of the same argument is still invalid.
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